Peter Selinger- Semantics of a Quantum Programming Language

8/3/2019 Peter Selinger- Semantics of a Quantum Programming Language

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Semantics of a Quantum Programming Language

Peter Selinger

Dalhousie University

Halifax, Canada

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Why Quantum Programming Languages?

For certain problems, quantum algorithms have anexponential speedup over best known classical algorithms.

Most research in quantum computing has focused onalgorithms and complexity theory.

Quantum algorithms are traditionally described in terms ofhardware: quantum circuits or quantum Turing machines.

Want compositionality. Also, how do quantum featuresinteract with other language features such as structured

data, recursion, i/o, higher-order.

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Part I: Quantum Computation

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The QRAM abstract machine [Knill96]

Classical device:

master

(general purpose)

Quantum device:

slave

control

results

General-purpose classical computer controls a specialquantum hardware device

Quantum device provides a bank of individually addressablequbits.

Left-to-right: instructions.

Right-to-left: results.4


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Linear Algebra Review

Scalars C, column vectors u Cn, matrices A Cnm.

Adjoint A = (aji)ij, trace tr A = iaii, norm

A2 = ij |aij|2. Unitary matrix S Cnn if SS = I.

Change of basis: B = SAS

tr B = tr A, B = A.

Hermitian matrix A Cnn: if A = A.Hermitian positive: uAu 0 for all u Cn.Diagonalization: A = SDS, S unitary, D real diagonal.

Tensor product A B, e.g.

0 1

1 0

B =

0 B

B 0

.

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Quantum computation: States

state of one qubit: |0

+ |1

(superposition of |0

and |1

).

state of two qubits: |00 + |01 +|10 + |11.

independent: (a|0 + b|1) (c|0 + d|1) =ac|00 + ad|01 + bc|10 + bd|11.

otherwise entangled.

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Lexicographic convention

Identify the basis states |00, |01, |10, |11 with the standardbasis vectors

1

0

0

0

,

0

1

0

0

,

0

0

1

0

,

0

0

0

1

,

in the lexicographic order.

Note: we use column vectors for states.

= |00 + |01 +|10 + |11.

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Quantum computation: Operations

unitary transformation

measurement

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Some standard unitary gates

Unary: Binary:

N =

0 11 0

, Nc =

I 00 N

,

H = 12

1 1

1

1, Hc =

I 0

0 H,

V =

1 0

0 i

, Vc =

I 0

0 V

,

W =

1 0

0

i

, Wc =

I 0

0 W

, X =

1 0 0 0

0 0 1 0

0 1 0 0

0 0 0 1

.

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Measurement

|0 + |1

||20

||21

|0 |1

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Two Measurements|00 + |01 +|10 + |11

||2+||2

0

||2+||2

1

|00 + |01

||2

||2+||20 ||2

||2+||21

|10 + |11

||2

||2+||20 ||2

||2+||21

|00 |01 |10 |11

Note: Normalization convention.

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Pure vs. mixed states

A mixed state is a (classical) probability distribution on

quantum states.

Ad hoc notation:

1

2

+

1

2

Note: A mixed state is a description of our knowledge of a

state. An actual closed quantum system is always in a (possiblyunknown) pure state.

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Density matrices (von Neumann)

Represent the pure state v =

C2 by the matrix

vv

=

C2

2.

Represent the mixed state 1 {v1} + . . . + n {vn} by

1v1v1 + . . . + nvnv

n.

This representation is not one-to-one, e.g.

1

2

1

0

+

1

2

0

1

=

1

2

1 0

0 0

+

1

2

0 0

0 1

=

.5 0

0 .5

1

2

12

1

1

+

1

2

12

1

1

=

1

2

.5 .5

.5 .5

+

1

2

.5 .5

.5 .5

=

.5 0

0 .5

But these two mixed states are indistinguishable.

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Quantum operations on density matrices

Unitary:

v

Uv vv

UvvU A

UAU

Measurement:

||20

||21

0

0

0

1

0

0 0

0 00

a bc d

a

0

d

1

a 00 0

0 00 d

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A complete partial order of density matrices

Let Dn = {A Cnn | A is positive hermitian and tr A 1}.

Definition. We write A B if B A is positive.

Theorem. The density matrices form a complete partial order

under

.

A A A B and B A

A = B

A

B and B

C A C every increasing sequence A1 A2 . . . has a least upper

bound

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Part II: The Flow Chart Language

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Earlier Quantum Programming Languages

Knill (1996): conventions for writing pseudo-code

Omer (1998): scratch space management, user definedoperators

Sanders and Zuliani (2000): specification language,

stepwise refinement

Bettelli, Calarco, and Serafini (2001): based on C++

Imperative languages, run-time checks and errors, no formal

semantics.

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A simple classical flow chart

input b, c : bit

branch b

0 1b, c : bit

b, c : bit

b, c : bit

b := c

b, c : bit

c := 0

b, c : bit

b, c : bitoutput b, c : bit

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Classical flow chart, with boolean variables expanded

00 01 10 11

00 01 10 11

( branch b )

( b := c )

( c := 0 )

( merge )

input b, c : bit

output b, c : bit

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Classical flow chart, with boolean variables expanded

00 01 10 11

00 01 10 11

A B C D

A B 0 0

0 0 C D

C 0 0 D

C 0 D 0

B D 0A + C

( branch b )

( b := c )

( c := 0 )

( merge )

input b, c : bit

output b, c : bit

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input b, c : bit

branch b

0 1b, c : bit

b, c : bit

b, c : bit

b := c

b, c : bit

c := 0

b, c : bit

b, c : bitoutput b, c : bit

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input b, c : bit

branch b

0 1 b, c : bit = (0,0,C,D)

b, c : bit = (A,B,C,D)

b, c : bit = (A,B,0,0)

b := c

b, c : bit = (C,0,0,D)

c := 0

b, c : bit = (C,0,D,0)

b, c : bit = (A + C,B,D,0)output b, c : bit

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Summary of classical flow chart componentsAllocate bit:

= A

new bit b := 0

b : bit, = (A, 0)

Discard bit:

b : bit, = (A, B)

discard b

= A + B

Assignment:

b : bit, = (A, B)

b := 0

b : bit, = (A + B, 0)

b : bit, = (A, B)

b := 1

b : bit, = (0, A + B)

Branching:

branch b

0b : bit, = (A, 0) 1b : bit, = (0, B)

b : bit, = (A, B)

Merge: Initial:

= A = B

= A + B

= 0

Permutation:

b1, . . . , bn : bit = A0, . . . , A2n1

permute

b(1), . . . , b(n) : bit = A2(0), . . . , A2(2n1)

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A simple quantum flow chart

input p, q : qbit

measure p

0

p,q : qbit1

p,q : qbit

p,q : qbit

q = Np,q : qbit

p = Np,q : qbit

p,q : qbit

output p, q : qbit

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A simple quantum flow chart

input p, q : qbit

measure p

0

p,q : qbit =

A 0

0 0

1

p,q : qbit =

0 0

0 D

p,q : qbit =

A B

C D

q = N

p,q : qbit =

NAN 0

0 0

p = Np,q : qbit =

D 0

0 0

p,q : qbit =

NAN + D 00 0

output p, q : qbit

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Summary of quantum flow chart components

Allocate qbit:

= A

new qbit q := 0

q : qbit, = A 00 0

Discard qbit:

q : qbit, =

A BC D

discard q

= A + D

Unitary transformation:

q : qbit, = A

q = S

q : qbit, = (S I)A(S I)

Measurement:

measure q

0q : qbit, =

A 00 0

1q : qbit, =

0 00 D

q : qbit, =

A BC D

Merge: Initial:

= A = B

= A + B

= 0

Permutation:

q1

, . . . , qn

: qbit = (aij

)ij

permute

q(1), . . . , q(n) : qbit = (a2(i),2(j))ij

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Combining classical data with quantum data

Consider typing contexts of the form

b1 : bit, . . . , bn : bit, q1 : qbit, . . . , qm : qbit.

Definition. A state for the above typing context is a 2n-tuple

(A0, . . . , A2n1) of density matrices, each of dimension 2m

2m.

tr (A0, . . . , A2n1) :=

i tr Ai,

(A0

, . . . , A2

n1

)

:= (A0

, . . . , A2

n1

),

S(A0, . . . , A2n1)S := (SA0S, . . . , SA2n1S),

|(A0, . . . , A2n1)|2 :=

i |Ai|

2.

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Summary of language features:

our language is functional (no side effects) and staticallytyped (no run-time errors).

it combines quantum and classical features (the compilercan separate them again).

it has high-level features (such as loops, recursion, andstructured data types) [not shown in this talk]

there is a compositional denotational semantics.

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The denotation of a quantum flow chart

The denotation of a flow chart is a function which maps

(tuples of) matrices to (tuples of) matrices.

Example: the denotation of the quantum flow chart from

previous slide is the function

F

A B

C D

=

NAN + D 0

0 0

.

Question: Which functions can occur?

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Superoperators

1) linear

2) positive: A positive F(A) positive

3) trace non-increasing: A positive tr F(A) tr(A)4) completely positive: F idn positive for all n

Theorem: The above conditions are necessary and sufficient.

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The category Q of superoperators

Objects: signatures = n1, . . . , nk

Morphisms: f : is a superoperatorf : Cn1n1 . . . Cnknk

Cm1m1 . . . Cmkmk

Structure:

symmetric monoidal category (horiz.+vert. composition) coproducts (merge, initial)

CPO-enriched (fixpoints, recursion) traced monoidal (loops)

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Structural and denotational equivalence

Definition. An elementary quantum flow chart category is a

symmetric monoidal category with traced finite coproducts,

such that A () is a traced monoidal functor for every objectA, together with a distinguished object qbit and morphisms

: I I qbit and : qbit I I, such that = id.Definition. Two quantum flow charts X, Y are structurally

equivalent if for every elementary quantum flow chart categoryC and every interpretation of basic operator symbols,

[[X]] = [[Y]].

We say X and Y are denotationally equivalent if [[X]] = [[Y]] for the

canonical interpretation in the category Q of signatures and

superoperators.

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Higher-order quantum computation

Consider functions of higher-order types such as(qbit bit) qbit etc.

Quantum data is subject to linearity constraints. Need to

avoid terms that lead to runtime errors such aslet q = new qbit() in (x.H(x, x))q.

Bits are always duplicable, qubits are never duplicable.

What about functions?

Considerq:qbit p.p : qbit

qbit

q:qbit p.q : qbit qbitBoth closures have type qbit qbit, but only the first oneis duplicable.

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Linear type system [Selinger,Valiron04]

Types: A, B ::= !A A B 1 A B.

Subtyping: !A


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Example: quantum teleportation / dense coding

It is possible to write a function

TPPair : 1 (qbit bit bit) (bit bit qbit)such that for any application

(f, g) = TPPair(),

one obtains a pair of functions (f, g) with the properties:

f : qbit bit bitg : bit

bit

qbit

f g = idbitbitg f = idqbit

Thus, are the types qbit and bit bit isomorphic?

Answer: No, because each such pair f, g can only be used once.

Thus we have a single-use type isomorphism. This is a

curious phenomenon.

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Overview of some recent research

Quantum process calculi. Lalire-Jorrand (2004),Gay-Nagarajan (2004), Adao-Mateus (2005)

Higher-order quantum computation. Van Tonder (2003,2004), Selinger-Valiron (2004), Altenkirch-Grattage (2004)

Categorical quantum computation. Abramsky-Coecke(2004), Selinger (2005)

Measurement based quantum computation.Danos-DHondt-Kashefi-Panangaden (2004, 2005)

Quantum specification. Zuliani (2001-2004),DHondt-Panangaden (2004), Tafliovich (2004)

Quantum coherent spaces. Girard (2003), Selinger(2004)

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Peter Selinger- Semantics of a Quantum Programming Language

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